We prove the $H^{infty}$ well-posedness of the forward Cauchy problem for a $Psi$-do differential operator $P$ of order $mgeq 2$ with Log-Lipschitz continuous symbol in the time variable. The characteristic roots $lambda_k$ of $P$ are distinct and satisfy the necessary Lax-Mizohata condition Im$lambda_kgeq 0$. The Log-Lipschitz regularity has been tested as the optimal one for $H^{infty}$ well-posedness in the case of second order hyperbolic operators. Our main aim is to present a simple proof which needs only a little of the basic calculus of standard $Psi$-do differential operators.
M. Cicognani, R. Agliardi (2004). The Cauchy problem for a class of Kovalevskian pseudo-differential operators. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 132, 841-845 [10.1090/S0002-9939-03-07092-8].
The Cauchy problem for a class of Kovalevskian pseudo-differential operators.
CICOGNANI, MASSIMO;AGLIARDI, ROSSELLA
2004
Abstract
We prove the $H^{infty}$ well-posedness of the forward Cauchy problem for a $Psi$-do differential operator $P$ of order $mgeq 2$ with Log-Lipschitz continuous symbol in the time variable. The characteristic roots $lambda_k$ of $P$ are distinct and satisfy the necessary Lax-Mizohata condition Im$lambda_kgeq 0$. The Log-Lipschitz regularity has been tested as the optimal one for $H^{infty}$ well-posedness in the case of second order hyperbolic operators. Our main aim is to present a simple proof which needs only a little of the basic calculus of standard $Psi$-do differential operators.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
